In this topic, you study the Fourier Transform Properties as Linearity, Time Scaling, Time Shifting, Frequency Shifting, Time differentiation, Time integration, Frequency differentiation, Time Reversal, Duality, Convolution in time and Convolution in frequency.
Linearity
if
\[a{f_1}(t) \leftrightarrow a{F_1}(\omega )\]
\[b{f_2}(t) \leftrightarrow b{F_2}(\omega )\]
Then
\[a{f_1}(t){\text{ }}+{\text{ }}b{f_2}(t) \leftrightarrow a{F_1}(\omega ){\text{ }} + {\text{ }}b{F_2}(\omega )\]
Time Scaling
if
\[f(t) \leftrightarrow F(\omega )\]
Then
\[f(at) \leftrightarrow \frac{1}{{\left| a \right|}}F\left( {\frac{\omega }{a}} \right)\]
Time Shifting
if
\[f(t) \leftrightarrow F(\omega )\]
Then
\[f(t – {t_0}) \leftrightarrow F({\omega _0}){e^{ – j\omega {t_0}}}\]
Frequency Shifting
if
\[f(t) \leftrightarrow F(\omega )\]
Then
\[f(t){e^{j{\omega _0}t}} \leftrightarrow F(\omega – {\omega _0})\]
Time differentiation
if
\[f(t) \leftrightarrow F(\omega )\]
Then
\[\frac{d}{{dt}}\left[ {f(t)} \right] \leftrightarrow j\omega F(\omega )\]
Time integration
if
\[f(t) \leftrightarrow F(\omega )\]
Then
\[\int\limits_{ – \infty }^t {f(\tau )d\tau } \leftrightarrow \frac{1}{{j\omega }}F(\omega )\]
Frequency differentiation
if
\[f(t) \leftrightarrow F(\omega )\]
Then
\[{t^n}f(t) \leftrightarrow {(j)^n}\frac{{{d^n}}}{{d{\omega ^n}}}F(\omega )\]
Time Reversal
if
\[f(t) \leftrightarrow F(\omega )\]
Then
\[f( – t) \leftrightarrow F( – \omega )\]
Duality
if
\[f(t) \leftrightarrow F(\omega )\]
Then
\[F(t) \leftrightarrow 2\pi f( – \omega )\]
Convolution in time
if
\[{f_1}(t) \leftrightarrow {F_1}(\omega )\]
\[{f_2}(t) \leftrightarrow {F_2}(\omega )\]
Then
\[{f_1}(t)*{f_2}(t) \leftrightarrow {F_1}(\omega ){F_2}(\omega )\]
Convolution in frequency
if
\[{f_1}(t) \leftrightarrow {F_1}(\omega )\]
\[{f_2}(t) \leftrightarrow {F_2}(\omega )\]
Then
\[{f_1}(t){f_2}(t) \leftrightarrow \frac{1}{{2\pi }}\left[ {{F_1}(\omega )*{F_2}(\omega )} \right]\]